In known manner, a fluid dispenser pump generally includes a pump body in which a piston is mounted to slide between a rest position and a dispensing position. When the pump is a metered dose pump, i.e. a pump that dispenses accurate and identical metered quantities or “doses” each time it is actuated, the pump body can contain a pump chamber that is generally defined between an inlet valve and an outlet valve. In which case, it is essential for the outlet valve to be leaktight during actuation so as to avoid any leakage before the end of actuation, and, similarly, the leaktightness of the inlet valve must also be excellent in order to avoid any of the fluid contained in the pump chamber flowing back into the reservoir during actuation. Another problem that can arise relates to priming, i.e. to removing the air contained in the pump chamber before the pump is actuated for the first time. In certain pumps, the air is removed via the outlet valve towards the dispensing orifice. That solution is not always su-table when the device is provided with closure means for closing off the dispensing orifice, and in particular when such closure means prevent the air from being removed during priming. Another solution consists in providing a hole in the pump body and in removing the air through said hole. Unfortunately, that solution makes it necessary to provide means for closing off the hole after priming, which can make the structure of the pump relatively complex.